Probability Langmuir-Hinshelwood based CO2 photoreduction kinetic models

For engineering solutions, scaling photoreactors and processes, kinetic models that describe the impact of process conditions on CO2 photoreduction are critical to driving this technology forward. Probability LangmuirHinshelwood based CO2 photoreduction kinetic models were developed after several criteria that included: a high purity photodifferential photoreactor with a high ratio of reagent gas volume to irradiated photocatalyst surface area and automated robust data collection and kinetic modelling using a MATLAB programme. Product distribution profiles indicated the dynamic changes occurring over the photocatalyst with an initial increase in H2 product distribution, followed by an increase in CH4 and finally CO product distribution, possibly due to the photocatalytic degradation of CH2O and CH2O2 intermediates. Production of H2 increased with a decrease in CH4 when the partial pressure of H2O was increased. Using the glyoxal mechanism, this is possibly explained via the formation of CH3CO2H from H2O reacting with CH3CHO that prevents the full conversion of CH3CHO to CH4. To account for deactivation, probability Langmuir–Hinshelwood based kinetic models were used to fit CO2 photoreduction kinetic data for CH4, CO and H2 with low average standard errors of × × − − 3.44 10 , 1.54 10 4 4 and × − 1.36 10 4, respectively. The probability LH based kinetic model coefficients were estimated with low standard deviations, using a robust and repeatable numerical method using a trust-region and multi-start algorithm. The models were used to predict optimised selectivity of CH4, CO and H2.


Introduction
CO 2 photoreduction offers a potential solution to minimising the impacts from rising anthropogenic CO 2 concentration [1]. The use of relatively abundant CO 2 also offers an opportunity for industries to be created through the production of value added chemicals [2]. CO 2 photoreduction offers one such avenue for the production of solar fuels. CO 2 photoreduction also offers the potential to produce H 2 and CO syngas that can be converted into higher order hydrocarbon fuels using well established Fisher-Tropsh chemistry.
Low CO 2 photoreduction efficiency towards products such as CH 4 , CO and H 2 has driven the focus on developing novel photocatalysts [3][4][5][6], with limited focus on understanding the CO 2 photoreduction mechanism [7,8] and the kinetics [9][10][11][12][13][14][15][16][17]. Standardisation of the practical aspects for collecting CO 2 photoreduction data is still under development. The impact of adventitious carbon from photocatalyst preparation, coating and photoreactor components on false positive production is a challenge for collecting CO 2 photoreduction data [18,19]. This challenge is compounded by not being able to quantitatively compare CO 2 photoreduction photocatalysts due to the wide variation in experimental protocols and photoreactors used [20]. In addition, analytical methods for tracking the reaction are not consistent with different purities of analytical gases and combination of detectors used [21]. For developing intrinsic CO 2 photoreduction kinetic models, the use of a photodifferential photoreactor that increases the probability of equal participation of illuminated active sites has not been addressed for developing CO 2 photoreduction kinetic models [22]. Deactivation of the CO 2 photoreduction photocatalyst is a challenge that has been reported by a growing number of authors [7]. The current Langmuir-Hinshelwood (LH) based kinetic models assume that the availability of active sites is constant and does not account for the deactivation of the photocatalyst. In addition, previous LH based CO 2 photoreduction kinetic models have included fractional coverage of H 2 O and CO 2 without considering the mole equivalents of a proposed elementary surface reaction. Numerical analysis has so far not been considered for evaluating the estimation of the CO 2 photoreduction kinetic model coefficients. This is a critical aspect as conclusions about the reaction parameters are made based on the coefficient values estimated. Nonlinear functions are often nonconvex with a large number of local minimum leading to a range of solutions and different coefficient values [23].
With the above challenges identified for developing intrinsic CO 2 photoreduction kinetic models, the five considerations for developing these models is described by the decision tree and was used to guide the kinetic model development presented in this work (Fig. 1).
In this work, to limit the impact of adventitious carbon, slurry deposition of commercial P25 TiO 2 with H 2 O onto stainless steel supports was tested in a chemically cleaned stainless steel based photoreactor setup ( Fig. 1 Steps 1-2). To increase the probability of active site participation leading to an intrinsic kinetic model, a photodifferential photoreactor with a high ratio of reactor gas volume to illuminated photocatalyst area was employed ( Fig. 1 Step 2). A robust gas chromatography analytical method using a combination of flame ionisation and thermal conductivity detectors, methaniser and variable valve timing was used for linear calibration for the expected CO 2 photoreduction products ( Fig. 1 Step 3). To account for deactivation, using a novel probability approach, a probability LH based kinetic model was used to account for the loss of active sites during the reaction towards CH 4 , CO and H 2 ( Fig. 1 Step 4). This probability approach yielded a different explanation to the current LH based kinetic models for the change in fractional coverage and the loss of active sites during CO 2 photoreduction over time. A Weibull probability density function (PDF) was re-parametrised to include active illuminated sites and deactivation over time with a LH based model. Elementary surface reactions towards CH 4 , CO and H 2 are used to account for the mole equivalents of the reaction in the kinetic model. A robust mean median multi-start trustregion numerical method is used to estimate the coefficients of the nonlinear probability LH based kinetic models ( Fig. 1 Step 5).

Catalyst coating
Stainless steel 80 gauge mesh (25 × 40 mm) was cleaned by sonication in 100 ml 1:9 70 % HNO 3 in DI H 2 O, neutralized in 100 ml 0.6 M Na 2 CO 3 in DI H 2 O, washed with 3 × 20 ml DI H 2 O and then dried at 80°C for 12 h. The mesh was coated with a slurry of 4.7 g P25 TiO 2 (Evonik) in 12 ml DI H 2 O. The slurry was covered and agitated in an ultrasound bath set at 50°C for 60 min. The stainless steel mesh was sonicated in the catalyst slurry for 30 s before being dried at 140°C for 3 h.

CO 2 photoreduction rate data
All CO 2 photoreduction experiments were performed under continuous flow conditions (Fig. 2).
The coated stainless steel mesh was placed in the middle of the photoreactor and sealed. Residual air in the system was evacuated via three repetitive steps of placing the system under vacuum to −1 bar and the vacuum released with CO 2 (99.995%) to +1 bar. The CO 2 was evacuated through the GC lines until approximately 0.2 bar, before the system was sealed and placed under vacuum again. The flow rate of CO 2 and Ar was set according to the experimental settings required and passed through the temperature controlled ( ± 0.1°C) aluminium body saturator for at least 12 h to allow the system to equilibrate. The total flow of the reagent gas was set to 0.35  information for the flow rates of CO 2 and Ar used. Relative humidity ( ± 1.8% RH) was measured using an inline Sensirion SHT75 humidity sensor potted (MG Chemicals 832HD) into a Swagelok 1/4" T-piece. The temperature of the photoreactor was controlled using a hotplate and the surface of the coated photocatalyst measured using a Radley's pyrometer ( ± 2.0°C). To prevent condensation at higher saturation temperatures, the lines from the outlet of the saturator up until the inlet of the H 2 O trap were heated and temperature controlled ( ± 0.1°C) with a heating rope and thermocouple (Fig. 2). An OmniCure S2000 fitted with a 365 nm filter was used as the light source. The light source was placed 30 mm above the surface of the coated meshes used. Please refer to Supplementary information for a photo of the light source setup. Irradiance at the exit of the fiber optic light guide was measured before each experiment using an OmniCure R2000 radiometer (± 5%).

Experimental space investigated
The work investigates the impact of photocatalyst fractional coverage on CO 2 photoreduction kinetics. To accomplish this, a wide experimental space of different partial pressures of CO 2 and H 2 O, was explored for collecting kinetic data for CO 2 photoreduction using P25 TiO 2 (Table 1) [9][10][11][12][13][14][15][16][17]. This is the first example describing CO 2 photoreduction kinetics using continuous experimental data points collected under continuous flow conditions. In addition, previous examples have not factored in the requirements of the decision tree proposed in this work for collecting CO 2 photoreduction kinetic data Fig. 1.

Limiting the impact of false positives on CO 2 photoreduction kinetic data
The following sections (Sections 3.2, 3.3, 3.4, 3.5) highlight our attempts at addressing the questions shown by the decision tree towards developing an intrinsic CO 2 photoreduction kinetic model ( Fig. 1 Steps 1-5). One of the most critical challenges that CO 2 photoreduction faces is the verification of the carbon source and limiting the impact of adventitious carbon on false positive production ( Fig. 1 Steps 1-2). In the absence of in situ techniques that include DRIFTs spectroscopy and using isotopically labelled 13 C, control experiments have been used to attempt to confirm the CO 2 as the carbon source. Control experiments used include a combination of experimental settings shown in Table 2.
By excluding an important component of the CO 2 photoreduction process, the bold text X ( Table 2) indicates what the control experiment is trying to test the impact of on the CO 2 photoreduction process. Although there are many reports of an insignificant amount of products produced under these conditions, there is a growing number of authors reporting significant to moderate levels of CO 2 photoreduction products when especially using the Type I control experiment (Table 2) [24,19,25,18,[26][27][28]    We performed Type I-IV controls with only the Type I control yielding CO 2 photoreduction products CH 4 , CO and H 2 . It is very difficult to completely remove all traces of the slow moving CO 2 gas when performing a Type I control experiment. Using a 3 × vacuum purge similar to the one described to prepare the CO 2 photoreduction test (Section 2.2) where Ar was used to release the vacuum instead of CO 2 and post purge flow rates of 15.13 − ml.min 1 Ar for 16 h yielded between 2530-7380 ppm CO 2 that remained in the system. Dilla reported maximum CO 2 photoreduction, using P25 TiO 2 , when using very low partial pressures of CO 2 with a maximum production reported when using 1000 ppm CO 2 [29]. Ideally, the amount of CO 2 should be zero to use a Type I control experiment as evidence for CO 2 being the carbon source for CO 2 photoreduction. Under atmospheric like conditions (400 ppm CO 2 ), CO 2 was shown to form a HCO 3 layer on TiO 2 (110) which was stable at temperatures below 400 K [30] HCO 3 is a potential intermediate for CO production [7] and the formation of this layer is very likely to occur during the preparation and storage of the photocatalyst [30]. In this work, we did not attempt to pre-clean the photocatalyst or subtract production under control Type I conditions as the deconvolution of the potential impact of adsorbed HCO 3 , adventitious carbon and CO 2 photoreduction at low partial pressures of CO 2 is too complex.
P25 TiO 2 was chosen as a photocatalyst for this study as it is commercially available and well studied [3,31,32]. P25 TiO 2 is very pure with limited adventitious carbon able to survive the oxidative AERO-XIDE® manufacturing process using temperatures of 1000-2400°C. To limit the impact of adventitious carbon, stainless steel mesh was used as a photocatalyst support. P25 TiO 2 was coated onto the stainless steel supports using a P25 TiO 2 / DI H 2 O slurry deposition. Coverage of the coating was estimated using Otsu thresholding as we have previously described (Fig. 3) [33]. A consistent pattern of cracks was observed when using a slurry deposition that yielded 12.80, 54.21 and 84.70 mg dry loading of P25 TiO 2 onto the mesh, with very similar coverage of 90.97, 90.06 and 90.47% respectively (Fig. 3). The area of the exposed photocatalyst on the coated meshes was independent of the dry loading mass of P25 TiO 2 and likely to yield an equal probability of illuminated surface area between the different meshes used. Therefore it is assumed that the Step 1 in Fig. 1 is met with P25 TiO 2 already scaled for production, a reproducible coating method with equal coverage and a simple scalable DI H 2 O slurry deposition method that limits the impact of adventitious carbon.

Analytical methods
Tracking the CO 2 photoreduction reaction by an inline GC minimises the impact of sample contamination and allows for a continuous flow CO 2 photoreduction process. For collecting accurate kinetic data, a linear calibration that includes the expected production range is critical. The CO and CH 4 kinetic data reported herein was interpolated from the calibration curves included in the Supplementary information. H 2 kinetic data and monitoring of O 2 and N 2 was extrapolated from the calibration curves (Supplementary information) due to the very low values observed and difficulty in calibration down to these levels with a loss of linearity. Air contamination is likely to impact the TCD area recorded for O 2 and N 2 , especially as the diluted calibration gas approaches the concentrations of O 2 and N 2 contamination from air during the calibration.

Photodifferential photoreactor
An intrinsic CO 2 photoreduction kinetic model ideally describes kinetic data that is not impacted by scale. To develop an intrinsic kinetic model, all active sites must ideally have the same probability of participating in the CO 2 photoreduction reaction. To achieve this, a photodifferential photoreactor that increases the probability of equal participation of active sites by uniform light distribution, is critical for recording kinetic data. The photoreactor used for collecting kinetic data  is shown in Fig. 4b. The photoreactor (h = 1 mm, r = 25 mm, V = 1.96 ml) was designed to increase the ratio of reagent gas volume to illuminated photocatalyst and encourage a uniform light distribution with the fiber optic light source placed directly above the quartz window of the photoreactor. Temperature is likely to impact the adsorption/desorption equilibrium of gases on the photocatalysts surface, the surface diffusion of the reagent gases H 2 O and CO 2 and possible intermediates that include: [7]. Temperature is also very likely to facilitate CO 2 photoreduction by overcoming thermal energy barriers [34]. Without consensus on the mechanism of CO 2 photoreduction and the possible intermediates, temperature is assumed to impact all photocatalytic reactions occurring on the surface of the photocatalyst. In this work, the impact of partial pressures of the reagent gases on the fractional coverage was investigated with temperature kept constant. The high heat capacity of the thick based (h = 2 cm) stainless steel photoreactor and measurement of the photocatalyst surface temperature with a pyrometer ensured high quality kinetic data was collected. This type of photoreactor increased the uniformity of light and heat distribution and improved the probability of the kinetic data recorded being an average that described the sum of equal opportunity of CO 2 photoreduction over the irradiated photocatalyst surface. Mass transfer restrictions that could potentially impact the kinetics of the CO 2 photoreduction reaction were possibly reduced by operating under continuous flow. Future work could include testing the impact of different flow rates on the kinetics observed in the photodifferential photoreactor presented in this work.
To maximise the saturation of reagent CO 2 gas with H 2 O, the CO 2 gas passed through a coil shaped copper tube in an aluminium body impinger with a PTFE insert (Fig. 4a). Saturation of CO 2 with H 2 O was measured using a low cost inline humidity sensor. The sensor recorded both % Rh and°C with small standard deviations of 0.21 for relative humidity measurements (%RH) and 0.08 for temperature measurements (°C). A representative example of the humidity sensors measurements and a more detailed setup description is shown in the Supplementary information.
Kinetic models can be impacted if the mass of photocatalyst is used to describe kinetic data in units of − − μmol.g . h cat 1 1 . Using mass of photocatalyst in the kinetic data units, the kinetic data is no longer a description of CO 2 photoreduction occurring at illuminated photocatalyst active sites but an average that includes the bulk of non-illuminated and non-participating photocatalyst sites inside the non illuminated photocatalyst. Using this kinetic data will yield an extrinsic kinetic model. With a photodifferential photoreactor, units of − − μmol.cm .h 2 1 that described the kinetic data over area illuminated were used in this study.
Assuming active sites are only on the photocatalyst surface, using area illuminated instead of the mass of the photocatalyst allowed for developing kinetic models that are scalable with respect to light and attempts to only include participating active sites. The coverage estimated ( Fig. 3) was used to calculate the kinetic data rates.

Probability Langmuir-Hinshelwood based kinetic model
The is the reaction order of light intensity (dimensionless); K i represent the equilibrium adsorption constants for reactants and products − P (bar ); i 1 refer to the partial pressures for reactants and products n (bar); indicates the number of reactants that are involved in the assumed surface reaction and; a i is the number of moles for each reactant from the assumed surface reaction and z indicates the number of all reactants and products According to the LH model Eq. (1), the rate of CO 2 photoreduction is directly proportional to the fractional coverage of the reagent gases. Under the assumption that all sites being equal and monolayer formation, the fractional coverage of the reagent gases involved in the elementary surface reaction should reach a steady state under continuous flow conditions in a photodifferential photoreactor at constant temperature and irradiance. However the LH based kinetic model does not account for deactivation of the photocatalyst, especially under continuous flow conditions, as described by a growing number of reports [36][37][38][39][40][41]33,[42][43][44]. To account for this deactivation, the change in production over time needs to be considered in the kinetic model along with a parameter that describes the deactivation leading to a decrease in active sites over time. The primary cause of deactivation is an ongoing investigation with evidence of the formation of bidentate carbonate species, peroxo and peroxocarbonate species on the photocatalysts surface possibly leading to deactivation [7,42]. One option to explain the deactivation over time is to assume elementary surface reactions that permanently deactivate the active sites. Integrating the rate of these surface deactivation reactions over time, allows to estimate the concentration of sites that deactivate over time and modify the fractional coverage. To account for the deactivation observed over time, a number of different derivatives of the LH based kinetic model Eq. (1) that included the concentration of deactivated sites over time have been developed and can be found in the Supplementary information. Initial attempts to fit the experimental data did not succeed due to lack of knowledge of the causes of deactivation.
Without consensus on the driving force behind the deactivation of the photocatalyst, a macroscopic view of the kinetic model is introduced here to describe this change with time.
All the phenomenon involved in the photocatalytic process can be approximated by the chance of photons reaching sites that remain active. Instead of assuming a rate of decay of active fractional coverage, we assume that the fractional coverage is constant and replace the irradiance term I ( ) α in Eq. (1) with a probability of photons as a probability density function (PDF) being able to reach the active sites. The probability decreases over time as in Eq.
where: t is time (s, min or h) and PDF is a probability density function A Weibull PDF was used to describe the deactivation of a Co containing catalysts for Fisher-Tropsh processes [45]. The two parameter Weibull PDF is shown by Eq. (3).
where PDF t ( ) is the probability density function; β is the shape parameter (Dimensionless); η is the scale parameter (Dimensionless) and t is time (h, s or min) The scale parameter η and shape parameter β have an impact on the shape and height of the Weibull PDF (Fig. 5 a), where increasing η broadens the Weibull PDF and increasing β increases the height of the Weibull PDF.
For modelling the deactivation of CO 2 photoreduction with a Weibull PDF, the Weibull PDF needs to be re-parametrised to include illuminated active sites and a deactivation term. The Weibull PDF scale parameter η is analogous to the impact of deactivation η ( ) d where a larger scale parameter increases the probability of photons reaching sites favouring a steady state of production.
The Weibull PDF shape parameter β term (Fig. 5b) is analogous to the impact of irradiance of illuminated sites successfully catalysing the reaction Eq. (4) where if more irradiance is successfully utilised in the CO 2 photoreduction process then a higher rate of production will be observed with the shape parameter β having an impact on the hight of the PDF and less of an impact on the shape of the PDF (Fig. 5b).
The two parameter Weibull function can be re-parametrised and used to describe the change in sites over time t ( ) shown by Eq. (5) which can then be used towards the probability LH based kinetic model Eq. (6). The inclusion of time and the flexible Weibull PDF allows for the dynamic kinetic modelling of CO 2 photoreduction over time which still includes a LH component.

Numerical methods for developing CO 2 photoreduction kinetic models
For the systematic and reproducible treatment of the CO 2 photoreduction kinetic data, a program ( Fig. 6) was developed in MATLAB to extract the GC data (GCIt), followed by a numerical method (numIt) to estimate the probability LH based kinetic model coefficients (Section 3.4). GCIt (Fig. 6) included importing raw GC data, fitting calibration curves, integrating peaks, calculating ppm and finally producing a matrix of the experimental settings (Irradiance, partial pressures of CO 2 and H 2 O and time) and a matrix of the CO 2 photoreduction kinetic data towards CH 4 , CO and H 2 . These two matrices were then used in the numIt programme for kinetic modelling. To find a global solution to the nonlinear probability LH based CO 2 photoreduction kinetic model, we developed a numerical method that estimated the coefficient value using the median value from the iterations of a trust-region and multistart numerical method. numIt (Fig. 6) iterated over a user selected number of iterations I ( ) U over a selected number of multi-start points MS ( ) U and trust-region method to estimate the kinetic model coefficients using the upper and lower constraints shown in Table 3. The choice of upper and lower constraints does impact the estimation of the nonlinear model coefficients. We initially attempted to use lower constraint = 0 and upper constraint = ∞, except for α with lower constraint = 0 and upper constraint = 1, but the numerical methods were unable to converge. To optimise computation time and to find a problem that converged, a lower constraint = 0 and upper constraint = 100 was chosen for k η K , ,

Probability LH based kinetic models
Using the experimental settings shown in Table 1, the production of CH 4 , CO or H 2 over time as a response and the numerical methods described in Section 3.5, the coefficients were estimated for three probability LH based kinetic models shown in Table 4.
The coefficients were estimated with small standard deviations indicating the reproducibility of the numerical methods outlined in Section 3.5 ( Table 5). The standard error of regression Eq. (7) was used to evaluate the probability LH based kinetic models ( Table 6). The standard error of regression (SE) was used as it gives a better idea of the differences, with units of − − μmol.cm . h 2 1 , between the experimental kinetic data and model used.
where SE is the standard error of regression of the probability LH based kinetic model  (Table 6). This is the first example of a CO 2 photoreduction kinetic model being fit to continuous data points collected for CH 4 , CO and H 2 over time. The three kinetic model coefficient values were estimated using all of the respective three hundred and fifty CH 4 , COand H 2 data points over time from the experiments shown in Table 1. This is important to note as the models are not the product of a curve fitting exercise where a single model is determined by fitting several experimental points over a range but fitted to a vector of data points over a varied experimental range that includes changes in both  Perfect line fits are not uncommon when fitting generalised models and this was highlighted by the less than perfect fits to the H 2 production data when changing the partial pressure of CO 2 (Fig. 7 c). The H 2 kinetic model was still able to account for differences in maximum production and also a delayed but similar trend to the experimental values. It is also important to note that H 2 production did not vary when changing the partial pressure of CO 2 and the generalised H 2 kinetic model may not have been able to account for these very slight variations.

Impact of partial pressure CO 2
Production of CO and CH 4 increased when lower partial pressure of CO 2 was used (Fig. 7). This has previously been explained by an increase in competition between CO 2 and H 2 O for active sites on the catalyst [10,12] or by a reduction in photogenerated charge carriers on the photocatalyst surface [29]. To investigate the impact of partial pressure of CO 2 , the product distribution trends of CH 4 , CO and H 2 over time were evaluated using Eq. (8).
where PD is the product distribution of the desired product as a fraction; mḊ is the molar flow rate of the desired product − (μmol. h ) 1 ; mȦ is the molar flow rate of desired and undesired products − (μmol. h ) 1 and n is the number of desired and undesired products.
During ≈ 0-1 h, the product distribution of CO decreased (Fig. 9 b) whilst the product distribution of H 2 and CH 4 increased (Fig. 9 c,a). During the initial reaction times (≈ 0-2 h), the product distribution towards H 2 was favoured over CH 4 (Fig. 9 c,a) with the product distribution of H 2 peaking at ≈ 1 h. As the reaction time progressed, product distribution towards CH 4 was favoured over H 2 with CH 4 product distribution peaking at ≈ 2-3 h (Fig. 9 a). CO 2 photoreduction towards CH 4 is an eight electron process ( Table 4) that very likely involves the formation of a number of reaction intermediates (Fig. 11) [7]. The product distribution of CH 4 (Fig. 9 a) may peak later due to the rate limitations of the reaction intermediates forming on the photocatalyst surface. Product distribution of CO showed an inverse relationship to H 2 product distribution (Fig. 9 b,c) with an initial decrease and lowest product distribution of CO at ≈ 1-1.5 h (Fig. 9 b). It is likely that during Table 4 Assumed surface reactions and the probability LH based models used in this work for the production of CH 4 , CO and H 2 . * Detailed derivation of the models described in this work are described in the Supplementary information.  Table 5 Coefficient values and their respective standard deviation for the probability LH based kinetic models Table 4   During the latter stages of the reaction ≈ − ( 2 5h), the increase in product distribution of CO and decrease in product distribution for H 2 and CH 4 is likely related to deactivation where active sites conducive to the production of H 2 and CH 4 decrease with time. Overall the product distribution profiles provide evidence for the following: initially, a layer of H 2 O dominates active sites due to increased hydrophilicity of TiO 2 when irradiated with UV-light [46]. During this time, H 2 O oxidation is favoured with production of H + and H 2 . H + is then used for CO 2 reduction towards surface intermediates and eventually to the delayed, relative to H 2 production, CH 4 production. The types of surface intermediates formed during this stage is very likely to include CH 2 O and CH 2 O 2 H [7]. Both CH 2 O and CH 2 O 2 are known to undergo photocatalytic degradation on TiO 2 leading to the production of CO during the latter stages of the CO 2 photoreduction reaction [47,48].
Coefficients α and η d were placed in the re-parametrised Weibull PDF Eq. (5) with impacts on the shape and height of the fitted kinetic models (Fig. 5). The shape of the curves for CH 4 , CO and H 2 production were all different and would impact the estimation of both the α and η d coefficients. Attempts at decoupling this possibility by placing the I α term outside the PDF term (Please see Supplementary information) were unsuccessful where the modified function was unstable and did not always converge to a solution.
Bigger η d coefficients would indicate less deactivation (Fig. 5). The production of CH 4 , CO and H 2 followed the trend of CO > CH 4 > H 2 in terms of stability (Fig. 7, 8). Stability was defined as the period for which the photocatalyst produced a particular compound before apparent deactivation causing a drop in production. This correlated with a decrease in the deactivation scale parameter η d where CO > CH 4 > H 2 ( Table 5). The α coefficient values estimated followed the trend of H 2 > CH 4 > CO with an inverse relationship to η d (Table 5). This is expected as increasing α values, at constant irradiance, also impact the shape of the model with increasing values yielding sharper peaks with larger heights but also with a greater deactivation trend (Fig. 5). Irradiance is possibly playing a role in the deactivation of the photocatalyst with different relative impacts on CH 4 , CO and H 2 production. This was reflected in the different α coefficients estimated for CH 4 , CO and H 2 (  [12,49]. Increasing partial pressure of H 2 O from 2.66, 3.47 and 6.64 kPa did not impact the production of CO greatly (Fig. 8). However, increasing the partial pressure of H 2 O increased the production of H 2 and decreased the production of CH 4 (Fig. 8). The trend  for deactivation (CO > CH 4 > H 2 ) and product distribution (Fig. 10) at different partial pressures of H 2 O was similar for the different partial pressures of CO 2 experiments (Section 3.7).
The decrease in CH 4 with increasing partial pressure of H 2 O could possibly be explained by an increase in competition for active sites, as explained in Section 3.7, between H 2 O and CO 2 [12,29]. In addition, another explanation exists using the gyloxal mechanism, which is a speculative mechanism for CO 2 photoreduction (Fig. 11) [7].
With the glyoxal mechanism in mind, the reactivity of the organic intermediate acetaldehyde (CH 3 CHO) (Fig. 11) can also be considered where the nucleophilic attack of H 2 O on the C = O group of CH 3 CHO towards acetic acid (CH 3 CO 2 H). The conversion of CH 3 CHO towards CH 3 CO 2 H has been shown to occur for the photocatalysis on TiO 2 [50].

Optimising selectivity
Using the probability LH based kinetic models (Table 4) and coefficients estimated (5), the selectivity shown in Table 7 was optimised using an interior-point algorithm (MATLAB 'fmincon' function) and constraints using the minimum and maximum experimental range used for time, partial pressure of H 2 O and CO 2 .
The optimised values (Table 7) matched the observations made about the product distributions where H 2 formation is favoured initially, followed by CH 4 and finally CO formation. To maximise CH 4 selectivity (Table 7), the reaction would need to be stopped at 2.69 h with maximum partial pressure of CO 2 (98.38 kPa) and at first glance a surprisingly a high partial pressure of H 2 O (5.44 kPa) as it would be expected more H 2 would be produced. Increasing the partial pressure of H 2 O increased the production of H 2 (Fig. 8 c) with a decrease observed for CO (Fig. 8 b). However, the production of H 2 is made less significant by the more steady state production of CO over time. CO selectivity was maximised (Table 7), depending on the lower constraints used (Table 8), at a extremely short reaction time (7.31 × − 10 17 h) or at the maximum time (5 h). CO production is likely slowed down by the    11. Potential glyoxal mechanism pathway to describe CH 4 production from CO 2 photoreduction [7]. occupation of active sites towards H 2 and the reaction intermediates towards CH 4 . Selectivity of H 2 was optimised after a reaction time of 1.57 h, maximum partial pressure of H 2 O and minimum partial pressure setting of CO 2 ( Table 8). The reaction towards H 2 is initially fast but does not last long.

Conclusion
Developing CO 2 photoreduction kinetic models is a challenge due to addressing the challenges of limiting the impact of adventitious carbon, tracking the extremely low production of products and developing kinetic models with uncertainty about the mechanism that include the formation of several products and deactivation of the photocatalyst. This is the first example that addresses these challenges for developing CO 2 photoreduction kinetic models.
A simply slurry deposition coating method, high purity setup with an inline humidity sensor to measure humidity, a photodifferential photoreactor with high reagent gas volume to photocatalyst area illuminated, a calibrated GC method for the quantification of O 2 , N 2 , CH 4 , CO and H 2 , a MATLAB programme for data collection and kinetic modelling was used to develop CO 2 photoreduction kinetic models.
To account for deactivation, a flexible Weibull PDF was combined with a LH based kinetic model to describe CO 2 photoreduction towards CH 4 , CO and H 2 for different partial pressures of CO 2 and H 2 O. Increasing the partial pressure of CO 2 decreased the production of CO and CH 4 and is most likely due to the competition between CO 2 and H 2 O for adsorption to active sites. Increasing the partial pressure of H 2 O increased H 2 production with a decrease in CH 4 production. Using the glyoxal mechanism, one plausible explanation is the formation of CH 3 CO 2 H from CH 3 CHO and H 2 O preventing the formation of CH 4 .
The probability LH based kinetic model generalised well for describing the kinetics of CO 2 photoreduction towards CH 4  Product distribution profiles of CH 4 , CO and H 2 showed that the CO 2 photoreduction reaction was dynamic with an initial increase in H 2 product distribution, followed by an increase in CH 4 and finally CO product distribution. These changes are likely the result of different processes being favoured on the photocatalyst surface over time where initially H 2 O oxidation and the formation of intermediates leading to CH 4 are favoured. Finally, CO formation is favoured due to the possible photocatalytic degradation of accumulated CH 2 O and CH 2 O 2 species on the photocatalyst surface. The probability LH based kinetic models predicted the optimisation of CH 4 , CO and H 2 selectivity for time, partial pressures of H 2 O and CO 2 that matched well with the experimental data.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Table 7 Optimised time, partial pressures of H 2 O and CO 2 for the desired selectivity using the probability LH based kinetic models (Table 5). * Lower time bound 0.5 h used (